Since the goal of DSP is usually to measure or filter continuous
real-world analog signals, the first step is usually to convert the
signal from an analog to a digital form, by using an analog-to-digital converter
(ADC). Often, the required output signal is another analog output
signal, which requires a digital-to-analog converter
(DAC). Even if this process is more complex than analog processing
and has a discrete value range, the stability of
digital signal processing thanks to error detection and
correction and being less vulnerable to noise makes it advantageous over analog signal
processing for many, though not all, applications.[1]

Contents

DSP
domains

In DSP, engineers usually study digital signals in one of the
following domains: time
domain (one-dimensional signals), spatial domain
(multidimensional signals), frequency domain, autocorrelation
domain, and wavelet domains.
They choose the domain in which to process a signal by making an
informed guess (or by trying different possibilities) as to which
domain best represents the essential characteristics of the signal.
A sequence of samples from a measuring device produces a time or
spatial domain representation, whereas a discrete Fourier transform
produces the frequency domain information, that is the frequency
spectrum. Autocorrelation is defined as the cross-correlation of the signal with
itself over varying intervals of time or space.

Signal
sampling

With the increasing use of computers the usage of and need for digital
signal processing has increased. In order to use an analog signal
on a computer it must be digitized with an analog-to-digital
converter. Sampling is usually carried out in two stages, discretization
and quantization. In the
discretization stage, the space of signals is partitioned into equivalence
classes and quantization is carried out by replacing the signal
with representative signal of the corresponding equivalence class.
In the quantization stage the representative signal values are
approximated by values from a finite set.

The Nyquist–Shannon sampling
theorem states that a signal can be exactly reconstructed from
its samples if the sampling
frequency is greater than twice the highest frequency of the
signal. In practice, the sampling frequency is often significantly
more than twice the required bandwidth.

A digital-to-analog converter is used to convert the digital
signal back to analog. The use of a digital computer is a key
ingredient in digital control systems.

Time and
space domains

The most common processing approach in the time or space domain
is enhancement of the input signal through a method called
filtering. Filtering generally consists of some transformation of a
number of surrounding samples around the current sample of the
input or output signal. There are various ways to characterize
filters; for example:

A "linear" filter is a linear
transformation of input samples; other filters are
"non-linear." Linear filters satisfy the superposition condition,
i.e. if an input is a weighted linear combination of different
signals, the output is an equally weighted linear combination of
the corresponding output signals.

A "causal" filter uses only previous samples of the input or
output signals; while a "non-causal" filter uses future input
samples. A non-causal filter can usually be changed into a causal
filter by adding a delay to it.

A "time-invariant" filter has constant properties over time;
other filters such as adaptive filters change in time.

Some filters are "stable", others are "unstable". A stable
filter produces an output that converges to a constant value with
time, or remains bounded within a finite interval. An unstable
filter can produce an output that grows without bounds, with
bounded or even zero input.

A "finite impulse response" (FIR) filter uses only the input
signal, while an "infinite impulse response" filter (IIR) uses both the input signal
and previous samples of the output signal. FIR filters are always
stable, while IIR filters may be unstable.

Frequency
domain

Signals are converted from time or space domain to the frequency
domain usually through the Fourier transform. The Fourier
transform converts the signal information to a magnitude and phase
component of each frequency. Often the Fourier transform is
converted to the power spectrum, which is the magnitude of each
frequency component squared.

The most common purpose for analysis of signals in the frequency
domain is analysis of signal properties. The engineer can study the
spectrum to determine which frequencies are present in the input
signal and which are missing.

Filtering, particularly in non-realtime work can also be
achieved by converting to the frequency domain, applying the filter
and then converting back to the time domain. This is a fast, O(n
log n) operation, and can give essentially any filter shape
including excellent approximations to brickwall
filters.

There are some commonly used frequency domain transformations.
For example, the cepstrum
converts a signal to the frequency domain through Fourier
transform, takes the logarithm, then applies another Fourier
transform. This emphasizes the frequency components with smaller
magnitude while retaining the order of magnitudes of frequency
components.

Frequency domain analysis is also called spectrum- or
spectral analysis.